Foliations with measure preserving holonomy

Abstract

It has been known for some time that qualitative results concerning flows in the plane do not fully generalize, either to flows on arbitrary manifolds or to foliations of codimension one. On the other hand, in [11] it was shown that the Poincare-Bendixson Theorem can be generalized to foliations of codimension one provided that the leaves of the foliation satisfy a growth condition. In the classical situation of flows in the plane this condition is trivially satisfied. If a foliation comes from a finite dimensional Lie group action then the growth condition may be interpreted in terms of the ergodic properties of the group. Of course, for arbitrary foliations we do not necessarily have such a group action, but we do have a pseudogroup, called the holonomy pseudogroup, which acts transversely to the leaves. Growth conditions on the leaves can be interpreted as conditions on the holonomy pseudogroup which, in some cases, imply the existence of a measure which is invariant under the action of the pseudogroup. It is such invariant measures that capture the essence of the classical qualitative theory of flows on surfaces. The basic idea of using invariant measures to study qualitative aspects of foliations goes back to the rotation numbers of Poincare. A modern treatment of these notions for flows on metric spaces which clearly brings out the role of invariant measures is in Schwartzman [22]. This last work is based on Kryloff-Bogoliuboff [7] in which the space of invariant measures of a dynamical system is studied at length and is recognized to be a topological invariant of the dynamical system itself. Invariant measures have also appeared earlier in the qualitative study of foliations. Sacksteder [21] uses the notion of an invariant measure to describe the structure of smooth codimension one foliations having trivial holonomy groups, and Sinai [23] considers invariant measures for the foliations invariant under a transitive Anosov diffeomorphism. In both cases the invariant measures involved were positive on open sets. Hirsch and Thurston [4] consider more general invariant measures for foliated bundles